24 
MR, A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
components occurs. Thus the constituent which represents a set has sub-constituents 
indicating all the correspondences of the components of the set. 
133. In a graphical representation of a set the diagram not only indicates what 
units, &c., are distinguished and what undistinguished, but is such that if we ignore 
geometrical position in accordance with sec. 41, and the reference letters in accordance 
with sec. 42, the units, &c., will be actually distinguished and undistinguished, and 
the diagram will represent the form of the set by being actually of that form. 
134. Any collection of units which is not a set may be said to he imperfect. 
135. If any collection of units is such that its component pairs are all distinguished 
from the pairs connecting it with units detached from it, the collection is a set. For 
let Imn . . . and pqr ... be any two undistinguished components of the collection, then 
if a be any unit of the collection, a unit b which is such that Imn ... a is undistin¬ 
guished from pqr ... 6 must also be a component of the collection, otherwise the 
pairs connecting b and p, q, r, . . . will be pairs connecting a unit not of the collection 
with units of the collection, and they will be undistinguished from the pairs connecting 
a and l, m, n, . . . which are all pairs of the collection. 
136. Let x be any unit of a single set Q of n units ; consider the pairs formed by 
x with other units of Q ; take any one of these xp; let the number of pairs xp, xq, xr, 
. . . which are undistinguished from xp, be m; then in the case of any other unit y of 
Q the number of pairs yi, yj, yk, . . . which are undistinguished from xp is m also. 
Further the number of pairs ax, bx, cx, . . . which are undistinguished from xp is also 
m; for the number m of such pairs must be the same in the case of each unit, and 
thus we have mn=m'n, i.e., m=m. 
Aspects unique with respect to Collections. 
137. If xyz . . . abc . . . >-< uvw . . . abc . . ., then the aspects xyz . . . and uvw 
may be said to be duplicates with respect to the collection a, b, c, . . . If there is no 
duplicate of xyz . . . with respect to a, b, c, . . . then xyz . . . may be said to be unique 
with respect to a, b, c, . . . It should be observed that if xyz ... is unique with 
respect to a, b,c,. . . tliere may be an aspect uvw . . . such that xyz. . . abc . . .>-< 
uvw. . . cab . . . 
138. If the aspect abc ... be unique with respect to the collection d, e,f . . . and if 
the apsect def. . . be unique with respect to the collection g, h, i, . . . ; then abc ... is 
unique with respect to g, h, i,. . . I give the proof in the case in which the three 
collections are detached ; the proof in other cases is somewhat longer but presents no 
difficulty. If abc ... is not unique with respect to g, h, i, . . . there are units p, q, r, 
. . . such that abc . . . ghi >- <pqr . . . glu . . ., and there are (sec. 94) units s, t, u, 
. . . such that abc . . . def. . . ghi . . . >-< pqr . . . stu . . . ghi . . . ; but def. . . 
is unique with respect to g, h, i, . . . therefore stu ... is def . . . , therefore abc . . . def 
